Symmetric monoidal noncommutative spectra, strongly self-absorbing $C^{\ast }$-algebras, and bivariant homology

Abstract

Continuing our project on noncommutative (stable) homotopy we construct symmetric monoidal $\infty$-categorical models for separable $C^{\ast }$-algebras $\mathtt{S}{\mathtt{C}}_{\infty }^{\ast }$ and noncommutative spectra $\mathtt{NSp}$ using the framework of Higher Algebra due to Lurie. We study smashing (co)localizations of $\mathtt{S}{\mathtt{C}}_{\infty }^{\ast }$ and $\mathtt{NSp}$ with respect to strongly self-absorbing $C^{\ast }$-algebras. We analyse the homotopy categories of the localizations of $\mathtt{S}{\mathtt{C}}_{\infty }^{\ast }$ and give universal characterizations thereof. We construct a stable $\infty$-categorical model for bivariant connective $\mathtt{E}$-theory and compute the connective $\mathtt{E}$-theory groups of ${\mathcal{O}}_{\infty }$-stable $C^{\ast }$-algebras. We also introduce and study the nonconnective version of Quillen's nonunital ${\mathtt{K}}^{\prime }$-theory in the framework of stable $\infty$-categories. This is done in order to promote our earlier result relating topological $\mathbb{T}$-duality to noncommutative motives to the $\infty$-categorical setup. Finally, we carry out some computations in the case of stable and ${\mathcal{O}}_{\infty }$-stable $C^{\ast }$-algebras.